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Start fractions: A unit for doing arithmatics with fractions.

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components/fractions/COPYING.LGPL.txt Normal file
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GNU LIBRARY GENERAL PUBLIC LICENSE
Version 2, June 1991
Copyright (C) 1991 Free Software Foundation, Inc.
59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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This is the file COPYING.modifiedLGPL.txt, it applies to several units in
the source-tree for fractions. See there if this
modification can be applied.
These files are distributed under the Library GNU General Public License
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<?xml version="1.0" encoding="UTF-8"?>
<CONFIG>
<ProjectOptions>
<Version Value="9"/>
<PathDelim Value="\"/>
<General>
<Flags>
<MainUnitHasCreateFormStatements Value="False"/>
<MainUnitHasTitleStatement Value="False"/>
</Flags>
<SessionStorage Value="InProjectDir"/>
<MainUnit Value="0"/>
<Title Value="fractest"/>
<UseAppBundle Value="False"/>
<ResourceType Value="res"/>
</General>
<i18n>
<EnableI18N LFM="False"/>
</i18n>
<VersionInfo>
<StringTable ProductVersion=""/>
</VersionInfo>
<BuildModes Count="2">
<Item1 Name="Debug" Default="True"/>
<Item2 Name="Release">
<CompilerOptions>
<Version Value="11"/>
<PathDelim Value="\"/>
<Target>
<Filename Value="fractest"/>
</Target>
<SearchPaths>
<IncludeFiles Value="$(ProjOutDir)"/>
<OtherUnitFiles Value="..\MijnLib"/>
<UnitOutputDirectory Value="lib\$(TargetCPU)-$(TargetOS)"/>
</SearchPaths>
<CodeGeneration>
<Optimizations>
<OptimizationLevel Value="2"/>
<VariablesInRegisters Value="True"/>
</Optimizations>
</CodeGeneration>
<Linking>
<Debugging>
<GenerateDebugInfo Value="False"/>
<UseLineInfoUnit Value="False"/>
<StripSymbols Value="True"/>
</Debugging>
<LinkSmart Value="True"/>
<Options>
<Win32>
<GraphicApplication Value="True"/>
</Win32>
</Options>
</Linking>
</CompilerOptions>
</Item2>
</BuildModes>
<PublishOptions>
<Version Value="2"/>
</PublishOptions>
<RunParams>
<local>
<FormatVersion Value="1"/>
</local>
</RunParams>
<Units Count="2">
<Unit0>
<Filename Value="fractest.lpr"/>
<IsPartOfProject Value="True"/>
</Unit0>
<Unit1>
<Filename Value="fractions.pp"/>
<IsPartOfProject Value="True"/>
<UnitName Value="fractions"/>
</Unit1>
</Units>
</ProjectOptions>
<CompilerOptions>
<Version Value="11"/>
<PathDelim Value="\"/>
<Target>
<Filename Value="fractest"/>
</Target>
<SearchPaths>
<IncludeFiles Value="$(ProjOutDir)"/>
<OtherUnitFiles Value="..\MijnLib"/>
<UnitOutputDirectory Value="lib\$(TargetCPU)-$(TargetOS)"/>
</SearchPaths>
<CodeGeneration>
<Checks>
<IOChecks Value="True"/>
<RangeChecks Value="True"/>
<OverflowChecks Value="True"/>
<StackChecks Value="True"/>
</Checks>
</CodeGeneration>
<Linking>
<Debugging>
<UseHeaptrc Value="True"/>
</Debugging>
</Linking>
</CompilerOptions>
<Debugging>
<Exceptions Count="3">
<Item1>
<Name Value="EAbort"/>
</Item1>
<Item2>
<Name Value="ECodetoolError"/>
</Item2>
<Item3>
<Name Value="EFOpenError"/>
</Item3>
</Exceptions>
</Debugging>
</CONFIG>

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program fractest;
{$mode objfpc}{$H+}
//{$MODESWITCH ADVANCEDRECORDS}
uses
Classes, sysutils, fractions, math;
var
F1, F2, Res: TFraction;
D: Extended;
Prec: Extended;
I64: Int64;
i: Integer;
G: Int64;
B: Boolean;
S: String;
I32: integer;
//R1,R2, D1, D2: Int64;
//StartMS, EndMS: Comp;
//i: Integer;
(*
function float2rat(x) {
var tolerance = 1.0E-6;
var h1=1; var h2=0;
var k1=0; var k2=1;
var b = x;
do {
var a = Math.floor(b);
var aux = h1; h1 = a*h1+h2; h2 = aux;
aux = k1; k1 = a*k1+k2; k2 = aux;
b = 1/(b-a);
} while (Math.abs(x-h1/k1) > x*tolerance); <<wrong must be "<"
return h1+"/"+k1;
}
*)
//uses method of Continued fractions
function FloatToFrac(F: Double; Precision: Double): TFraction;
var
H1, H2, K1, K2, A, tmp: Int64;
B, diff, test: Double;
begin
H1 := 1;
H2 := 0;
K1 := 0;
K2 := 1;
b := F;
repeat
A := Round(Floor(b));
tmp := H1;
H1 := (a * H1) + H2;
H2 := tmp;
tmp := K1;
K1 := (a * K1) + K2;
K2 := tmp;
B := 1 / (B - A);
test := H1 / K1;
diff := Abs(test - F);
until (diff < Precision) or (H1 > $B504F334) or (K1 > $B504F334); // $B504F334 = Sqrt(High(Int64))
Result.Numerator := H1;
Result.Denominator := K1;
end;
function SBT_FloatToFraction(AValue: Double; AMaxNumerator, AMaxDenominator: Int64): TFraction; overload;
// "Stern-Brocot-Tree"
// http://stackoverflow.com/questions/5124743/algorithm-for-simplifying-decimal-to-fractions
// Ported by "circular"
var
n: Integer;
lower_n, lower_d, upper_n, upper_d, middle_n, middle_d: Int64;
isNeg: Boolean;
backup, newResult: TFraction;
EPS: Double;
begin
EPS := 0.01/AMaxDenominator;
isNeg := AValue < 0;
if isNeg then
AValue := -AValue;
n := Trunc(AValue);
newResult := Fraction(Round(Avalue),1);
//newResult.Init(round(AValue),1);
if isNeg then newResult.Numerator := -newResult.Numerator;
backup := newResult;
AValue := AValue - n;
// Lower fraction is 0/1
lower_n := 0;
lower_d := 1;
// Upper fraction is 1/1
upper_n := 1;
upper_d := 1;
while true do
begin
if abs(newResult.Numerator/newResult.Denominator - n - AValue) <
abs(backup.Numerator/backup.Denominator - n - AValue) then
backup := newResult;
// middle fraction is (lower_n + upper_n) / (lower_d + upper_d)
middle_n := lower_n + upper_n;
middle_d := lower_d + upper_d;
newResult := fraction(n * middle_d + middle_n, middle_d);
newResult.Normalize;
if (newResult.Numerator > AMaxNumerator) or
(newResult.Denominator > AMaxDenominator) then
begin
result := backup;
exit;
end;
if isNeg then newResult.Numerator := -newResult.Numerator;
// AValue + EPS < middle
if middle_d * (AValue + EPS) < middle_n then
begin
// middle is our new upper
upper_n := middle_n;
upper_d := middle_d;
end else
// middle < AValue - EPS
if middle_n < (AValue - EPS) * middle_d then
begin
// moddle is our new lower
lower_n := middle_n;
lower_d := middle_d;
end else
// middle is our best fraction
begin
result := newResult;
exit;
end;
end;
end;
(*
typedef
struct{int num,denom} Fraction; //????
Fail:
D = 0.499938637949526
F.ToFloat = 0.500000000000000
Prec = 0.000000000000001
Diff = 0.000061362050474
Magnitude = 61362050474
F = 1/2
------
Fraction Fraction::toFract(double val)
{ // find nearest fraction
int intPart = (int)val;
val -= (double)intPart;
Fraction low(0, 1); // "A" = 0/1
Fraction high(1, 1); // "B" = 1/1
for (int i = 0; i < 100; ++i)
{
double testLow = low.denom * val - low.num;
double testHigh = high.num - high.denom * val;
if (testHigh < Precision * high.denom)
break; // high is answer
if (testLow < Precision * low.denom)
{ // low is answer
high = low;
break;
}
if (i & 1)
{ // odd step: add multiple of low to high
double test = testHigh / testLow;
int count = (int)test; // "N"
int num = (count + 1) * low.num + high.num;
int denom = (count + 1) * low.denom + high.denom;
if ((num > 0x8000) ||
(denom > 0x10000))
break;
high.num = num - low.num; // new "A"
high.denom = denom - low.denom;
low.num = num; // new "B"
low.denom = denom;
}
else
{ // even step: add multiple of high to low
double test = testLow / testHigh;
int count = (int)test; // "N"
int num = low.num + (count + 1) * high.num;
int denom = low.denom + (count + 1) * high.denom;
if ((num > 0x10000) ||
(denom > 0x10000))
break;
low.num = num - high.num; // new "A"
low.denom = denom - high.denom;
high.num = num; // new "B"
high.denom = denom;
}
}
return Fraction(intPart, 1) + high;
}
*)
Type
TFracFunc = function(D, Prec: Double): TFraction;
procedure FTF_Test(Func1, Func2: TFracFunc; Name: String);
var
F1,F2: TFraction;
D1, D2, Prec, Diff1, Diff2: Double;
i, Count: Integer;
T: TextFile;
procedure OpenT;
begin
if FileExists(Name+'_Test.txt') then
Append(T)
else
Rewrite(T);
end;
begin
AssignFile(T,Name+'_Test.txt');
OpenT;
writeln(T,'Testing: ',Name);
CloseFile(T);
// Randomize;
Count := 0;
repeat
D1 := Random;
Prec := 0.1;
for i := 1 to 15 do
begin
write('.');
try
F1 := Func1(D1, Prec);
Diff1 := Abs(F1.ToFloat - D1);
F2 := Func2(D1, Prec);
Diff2 := Abs(F2.ToFloat - D1);
if (Diff1 > Prec) or (Diff2 > Prec) then
begin
Inc(Count);
OpenT;
writeln(T,'Fail: ');
writeln(T,'D = ',D1:24:24);
writeln(T,'F1.ToFloat = ',F1.ToFloat:24:24);
writeln(T,'F2.ToFloat = ',F2.ToFloat:24:24);
writeln(T,'Prec = ',Prec:24:24);
writeln(T,'Diff1 = ',Diff1:24:24);
writeln(T,'Magnitude = ',Round(Diff1/Prec));
writeln(T,'Diff2 = ',Diff2:24:24);
writeln(T,'Magnitude = ',Round(Diff2/Prec));
writeln(T,'F1 = ',F1.ToString);
writeln(T,'F2 = ',F2.ToString);
CloseFile(T);
writeln;
writeln('Fail: ');
writeln('Fail: ');
writeln('D = ',D1:24:24);
writeln('F1.ToFloat = ',F1.ToFloat:24:24);
writeln('F2.ToFloat = ',F2.ToFloat:24:24);
writeln('Prec = ',Prec:24:24);
writeln('Diff1 = ',Diff1:24:24);
writeln('Magnitude = ',Round(Diff2/Prec));
writeln('Diff2 = ',Diff2:24:24);
writeln('Magnitude = ',Round(Diff2/Prec));
writeln('F1 = ',F1.ToString);
writeln('F2 = ',F2.ToString);
end;
except
on E: Exception do
begin
Inc(Count);
OpenT;
writeln(T,'Exception: ',E.Classname,', Message: ',E.Message);
writeln(T,' D=',D1:16:16,' Prec=',Prec:16:16);
CloseFile(T);
writeln;
writeln('Exception: ',E.Classname,', Message: ',E.Message);
writeln(' D=',D1:16:16,' Prec=',Prec:16:16);
end;
end;
Prec := Prec/10;
end;
until Count >= 15;
OpenT;
writeln(T,'Stopped after ',Count,' failures.');
CloseFile(T);
writeln('Stopped after ',Count,' failures.');
end;
procedure FTF_Test2(Func1, Func2: TFracFunc; Name: String);
var
F,F1,F2: TFraction;
D1, D2, Prec, Diff1, Diff2: Double;
A,B: Int64;
i, Count: Integer;
T: TextFile;
procedure OpenT;
begin
if FileExists(Name+'_Test2.txt') then
Append(T)
else
Rewrite(T);
end;
begin
AssignFile(T,Name+'_Test2.txt');
OpenT;
writeln(T,'Testing: ',Name);
CloseFile(T);
// Randomize;
Count := 0;
repeat
A := Random($8000000) + 1;
B := Random($8000000) + 1;
F.Numerator := Min(A,B);
F.Denominator := Max(A,B);
F.Normalize;
D1 := F.ToFloat;
Prec := 0.000000000000001;
for i := 1 to 1 do
begin
write('.');
try
F1 := Func1(D1, Prec);
F1.Normalize;
F2 := Func2(D1, Prec);
F2.Normalize;
if (F <> F1) or (F <> F2) then
begin
Inc(Count);
OpenT;
writeln(T,'Fail: ');
writeln(T,'D = ',D1:24:24);
writeln(T,'F1.ToFloat = ',F1.ToFloat:24:24);
writeln(T,'F2.ToFloat = ',F1.ToFloat:24:24);
writeln(T,'Prec = ',Prec:24:24);
writeln(T,'F = ',F.ToString);
writeln(T,'F1 = ',F1.ToString);
writeln(T,'F2 = ',F2.ToString);
writeln(T,'F<>F1 = ',F<>F1);
writeln(T,'F<>F2 = ',F<>F2);
CloseFile(T);
writeln;
writeln('Fail: ');
writeln('D = ',D1:24:24);
writeln('F1.ToFloat = ',F1.ToFloat:24:24);
writeln('F2.ToFloat = ',F1.ToFloat:24:24);
writeln('Prec = ',Prec:24:24);
writeln('F = ',F.ToString);
writeln('F1 = ',F1.ToString);
writeln('F2 = ',F2.ToString);
writeln('F<>F1 = ',F<>F1);
writeln('F<>F2 = ',F<>F2);
end;
except
on E: Exception do
begin
Inc(Count);
OpenT;
writeln(T,'Exception: ',E.Classname,', Message: ',E.Message);
writeln(T,' D=',D1:16:16,' Prec=',Prec:16:16);
CloseFile(T);
writeln;
writeln('Exception: ',E.Classname,', Message: ',E.Message);
writeln(' D=',D1:16:16,' Prec=',Prec:16:16);
end;
end;
Prec := Prec/10;
end;
until Count >= 15;
OpenT;
writeln(T,'Stopped after ',Count,' failures.');
CloseFile(T);
writeln('Stopped after ',Count,' failures.');
end;
(*
tofrac(dec) [dec is the (decimal) number to be converted]
{
num1 = 0 [these are integers]
den1 = 1
num2 = 1
den2 = 0
q = dec [q is a float, the current value being worked on]
n = 0
loop
{
n = n + 1
if (q > max_int) [prevent overflow]
exit loop
a = int(q) [a is an integer]
num = num1 + a * num2
den = den1 + a * den2
if (q - a < epsilon) [prevent divide by zero]
exit loop
q = 1 / (q - a)
num1 = num2
den1 = den2
num2 = num
den2 = den
}
until((abs(num/den - dec) < epsilon) [stop when close enough]
or
(n > max_steps) [avoid infinite loops]
or
(num > max_numerator) [stop if too big]
or
(den > max_denominator))
return num, den
}
*)
var
MaxDen,MaxNum: Int64;
Diff: Double;
begin
{
D = 0.478370306547731
F.ToFloat = 0.478370306564808
Prec = 0.000000000000001
Diff = 0.000000000017077
Magnitude = 17077
F = 135507/283268
}
F1 := Fraction(1,3);
F2 := Fraction(4,5);
Res := F1 * F2;
writeln(F1.ToString,' * ',F2.ToString,' = ',Res.ToString);
F1 := '1/15';
F2 := Res;
Res := F2 - F1;
writeln(F2.ToString,' - ',F1.ToString,' = ',Res.ToString);
F1 := Res;
Res := F1 + 2;
writeln(F1.ToString,' + 2 = ',Res.ToString,' which resolves to: ',Res.Resolve);
Res := FloatToFraction(0.5, 0.001);
writeln('FloatToFraction(0.5, 0.001) = ',Res.ToString);
F1 := Fraction(1,1,2);
writeln('Fraction(1,1,2) = ',F1.ToString,' = ',F1.Resolve);
F1 := Fraction(-1,1,2);
writeln('Fraction(-1,1,2) = ',F1.ToString,' = ',F1.Resolve);
F1 := Fraction(1,1,-2);
writeln('Fraction(1,1,-2) = ',F1.ToString,' = ',F1.Resolve);
F1 := Fraction(-1,1,-2);
writeln('Fraction(-1,1,-2) = ',F1.ToString,' = ',F1.Resolve);
try
F1 := Fraction(1,-1,2);
writeln('Fraction(1,-1,2) = ',F1.ToString,' = ',F1.Resolve,' SHOULD RAISE EXCEPTION!');
except
writeln('Fraction(1,-1,2): Exception raised, as expected');
end;
Exit;
D := 1E25;
Prec := 1;
B := TryFloatToFraction(D, Prec, F1);
writeln(D:16:16,', ',Prec:16:16,' : ',B,' Expected: FALSE');
//FloatToFraction := @CF_FloatToFraction;
D := 0.478370306547731;
Prec := 1E-19;
B := TryFloatToFraction(D, Prec, F1, True);
writeln(D:24:24,', ',Prec:24:24,' : ',B,' Expected: TRUE');
B := TryFloatToFraction(D, Prec, F1, False);
writeln(D:24:24,', ',Prec:24:24,' : ',B,' Expected: FALSE');
//if B then
begin
Diff := Abs(D-F1.ToFloat);
writeln('D = ',D:24:24);
writeln('F1 = ',F1.ToFloat:24:24);
writeln('Diff = ',Diff);
if (Diff>Prec) then writeln('Fail') else writeln('Ok');
end;
F2 := FloatToFractionDef(D,Prec, Fraction(1,1), False);
writeln('F2 = ',F2.ToString,', Expected: 1/1');
exit;
D := 0.478370306547731; //0.826824217801914;//0.235382874263451; //0.499938637949526;
Prec := 0.000000000000001 ; //0.000000000000001; //0.000000000001000; //0.000000000000001;
F1 := MF_FloatTofraction(D, Prec);
Diff := Abs(D-F1.ToFloat);
writeln('D = ',D:16:16);
writeln('F1 = ',F1.ToFloat:16:16);
writeln('Diff = ',Diff:16:16);
if (Diff>Prec) then writeln('Fail') else writeln('Ok');
//EXIT;
FTF_Test(@MF_FloatToFraction, {@}Fractions.FloatToFraction ,'MF-Def');
EXIT;
F1 := Fractions.FloatToFraction(10/3, 0.0000001);
writeln('F1 = ',F1.ToString);
exit;
//AllowZeroPowerZero := True;
//B := TryStrToint(paramstr(1), I32);
//if not B then I32 := 2;
//F1 := Fraction(0,2);
//F2 := F1 ** I32;
//writeln(F1.ToString,' ** ',I32,' = ',F2.ToString);
//EXIT;
//F1 := Fraction(5,13);
//F2 := Fraction(10,26);
//if not (F1 < F2) then write ('NOT '); writeln(F1.ToString ,' < ',F2.ToString);
//if not (F2 > F1) then write ('NOT '); writeln(F2.ToString ,' > ',F1.ToString);
//if not (F2 < F1) then write ('NOT '); writeln(F2.ToString ,' < ',F1.ToString);
//if not (F1 = F2) then write ('NOT '); writeln(F1.ToString ,' = ',F2.ToString);
//EXIT;
//FracSymbol := 'div';
{
S := ParamStr(1);
if S='' then S := '1 1'+FracSymbol+'2';
B := TryStrTofraction(S, F1);
write('B = ',B);
if B then writeln(' ',S, ' -> ',F1.ToString)
else writeln;
//if B then
//begin
// F2 := S;
// writeln('F1 = ',F1.ToString);
//end;
//F1 := StrToFractionDef(S, Fraction(123,456));
//writeln('F1 = ',F1.ToString);
exit;
}
D := 1.0 + 307/1700;
D := pi;
Prec := 0.01;
MaxDen := 10;
MaxNum := 4 * MaxDen ;
for i := 1 to 10 do
begin
writeln('Prec = ',Prec:16:16);
writeln('MaxNum = ',MaxNum,' MaxDen = ',MaxDen);
F1 := FloatTofraction(D, Prec);
writeln('FloatToFraction:');
writeln('Result = ',F1.ToString);
//writeln('D = ',D:16:16);
writeln('ToFloat = ',(F1.ToFloat):16:16);
writeln('Diff = ',Abs(D-(F1.ToFloat)):16:16);
F1 := SBT_FloatToFraction(D, MaxNum, MaxDen);
writeln('FloatToFraction2:');
writeln('Result = ',F1.ToString);
//writeln('D = ',D:16:16);
writeln('ToFloat = ',(F1.ToFloat):16:16);
writeln('Diff = ',Abs(D-(F1.ToFloat)):16:16);
Prec := Prec / 10;
MaxNum := 10*MaxNum;
MaxDen := 10*MaxDen;
writeln;
end;
EXIT;
F1 := Fraction(2,3);
F2 := Fraction(1,5);
Res := F1 + F2;
writeln(F1.ToString,' + ',F2.ToString,' = ',Res.ToString);
Res := F1 - F2;
writeln(F1.ToString,' - ',F2.ToString,' = ',Res.ToString);
Res := F1 * F2;
writeln(F1.ToString,' * ',F2.ToString,' = ',Res.ToString);
Res := F1 / F2;
writeln(F1.ToString,' / ',F2.ToString,' = ',Res.ToString);
Res := '123/456';
writeln('123/456 -> ',Res.ToString);
Res := '123';
writeln('123 -> ',Res.ToString);
Res := '456/123';
writeln('456/123: ToString = ',Res.ToString,' Resolve = ',Res.Resolve);
Res := '456/-123';
writeln('456/-123: ToString = ',Res.ToString,' Resolve = ',Res.Resolve);
Res := '25/5';
writeln('25/5: ToString = ',Res.ToString,' Resolve = ',Res.Resolve);
F1 := '10/20'; F2 := '2/4';
writeln('(',F1.ToString,' = ',F2.ToString,') = ',F1=F2);
F1 := '1/2'; F2 := '21/40';
writeln('(',F1.ToString,' = ',F2.ToString,') = ',F1=F2);
if not (F1 < F2) then write('NOT '); writeln(F1.ToString,' < ',F2.ToString);
if not (F2 > F2) then write('NOT '); writeln(F1.ToString,' > ',F2.ToString);;
if not (F1 <= F2) then write('NOT '); writeln(F1.ToString,' <= ',F2.ToString);;
if not (F1 >= F2) then write('NOT '); writeln(F1.ToString,' >= ',F2.ToString);;
F2 := -F1;
writeln('-F1 = ',F2.ToString);
end.

810
components/fractions/fractions.pp Normal file
View File

@@ -0,0 +1,810 @@
unit fractions;
{ A unit for doing arithmatics with fractions
Copyright (C) 2015 by Lazarus and FreePascal community
(http://www.lazarus.freepascal.org and http://www.freepascal.org)
Original code by
Bart Broersma (www.flyingsheep.nl)
HappyLarry
Portions copyright by David Peterson and The Math Forum @ Drexel,
(redistributed with the consent of the copyright holder)
This library is free software; you can redistribute it and/or modify it
under the terms of the GNU Library General Public License as published by
the Free Software Foundation; either version 2 of the License, or (at your
option) any later version with the following modification:
As a special exception, the copyright holders of this library give you
permission to link this library with independent modules to produce an
executable, regardless of the license terms of these independent modules,and
to copy and distribute the resulting executable under terms of your choice,
provided that you also meet, for each linked independent module, the terms
and conditions of the license of that module. An independent module is a
module which is not derived from or based on this library. If you modify
this library, you may extend this exception to your version of the library,
but you are not obligated to do so. If you do not wish to do so, delete this
exception statement from your version.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE. See the GNU Library General Public License
for more details.
You should have received a copy of the GNU Library General Public License
along with this library; if not, write to the Free Software Foundation,
Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
}
{$mode objfpc}{$H+}
{$MODESWITCH ADVANCEDRECORDS}
interface
uses
SysUtils, Math;
type
{ TFraction }
TFraction = record
private
FNumerator: Int64;
FDenominator: Int64;
procedure SetDominator(AValue: Int64);
public
procedure Normalize;
function ToString: String;
function Resolve: String;
function ToFloat: Double;
property Numerator: Int64 read FNumerator write FNumerator;
property Denominator: Int64 read FDenominator write SetDominator;
end;
TFloatToFractionFunc = function(Value, Precision: Double): TFraction;
TGreatestCommonDivisorFunc = function(a, b: Int64): Int64;
function Fraction(ANumerator, ADenominator: Int64): TFraction;
function Fraction(AIntPart, ANumerator, ADenominator: Int64): TFraction;
operator = (F1: TFraction; F2: TFraction) B: Boolean;
operator < (F1: TFraction; F2: TFraction) B: Boolean;
operator > (F1: TFraction; F2: TFraction) B: Boolean;
operator <= (F1: TFraction; F2: TFraction) B: Boolean;
operator >= (F1: TFraction; F2: TFraction) B: Boolean;
operator := (I: Int64) F: TFraction;
operator := (S: String) F: TFraction;
operator + (L: TFraction; R: TFraction) F: TFraction;
operator - (L: TFraction; R: TFraction) F: TFraction;
operator - (L: TFraction) F: TFraction;
operator * (L: TFraction; R: TFraction) F: TFraction;
operator * (L: TFraction; R: Int64) F: TFraction;
operator * (L: Int64; R: TFraction) F: TFraction;
operator / (L: TFraction; R: TFraction) F: TFraction;
operator / (L: TFraction; R: Int64) F: TFraction;
operator ** (L: TFraction; R: Integer) F: TFraction;
function GreatestCommonDivisor_DivisionBased(a, b: Int64): Int64;
function GreatestCommonDivisor_EuclidSubraction(a, b: Int64): Int64;
function CF_FloatToFraction(Value, Precision: Double): TFraction; //based upon Continue Fractios
function MF_FloatToFraction(Value, Precision: Double): TFraction; //Original code by David Peterson and The Math Forum @ Drexel
function StrToFraction(const S: String): TFraction;
function TryStrToFraction(const S: String; out F: TFraction): Boolean;
function StrToFractionDef(const S: String; Def: TFraction): TFraction;
function TryFloatToFraction(Value, Precision: Double; out F: TFraction; AcceptPrecisionError: Boolean = True): Boolean;
function FloatToFractionDef(Value, Precision: Double; Def: TFraction; AcceptPrecisionError: Boolean = True): TFraction;
function Min(a, b: TFraction): TFraction; inline; overload;
function Max(a, b: TFraction): TFraction; inline; overload;
function InRange(const AValue, AMin, AMax: TFraction): Boolean; inline; overload;
function EnsureRange(const AValue, AMin, AMax: TFraction): TFraction; inline; overload;
function Sign(const AValue: TFraction): TValueSign; inline; overload;
function IsZero(const AValue: TFraction): Boolean; overload;
function Abs(const AValue: TFraction): TFraction; overload;
function Floor(D: Double): Int64; overload;
procedure AdjustPrecision(var Precision: Double; Value: Double);
var
FloatToFraction: TFloatToFractionFunc = @MF_FloatTofraction;
GreatestCommonDivisor: TGreatestCommonDivisorFunc = @GreatestCommonDivisor_DivisionBased;
ResourceString
SDenominatorCannotBeZero = 'The denominator of a fraction cannot be 0';
const
FracSymbol: string = '/';
AllowZeroPowerZero: Boolean = False;
SDivisionByZero = 'Division by zero';
SZeroPowerZero = 'Raising 0 to the power 0 is undifined';
SInvalidFraction = '"%s" is not a valid fraction';
SRangeCheckError = 'Range check error';
const
MaxInt64 = High(Int64);
MinInt64 = Low(Int64);
implementation
//Math.Floor returns Integers and does NOT cause RangeCheck or OverFlow errors,
//So Math.Floor(1/9E-11) returns
// -1773790777 [FFFFFFFF964619C7] instead of
// 11111111111 [00000002964619C7] which is the correct Int64 answer
//this causes an infinite loop in CF_FloatToFraction function
function Floor(D: Double): Int64;
begin
Result := Trunc(D);
if Frac(D) < 0 then
Result := Result - 1;
end;
function GreatestCommonDivisor_DivisionBased(a, b: Int64): Int64;
var
temp: Int64;
begin
while b <> 0 do
begin
temp := b;
b := a mod b;
a := temp
end;
result := a
end;
function GreatestCommonDivisor_EuclidSubraction(a, b: Int64): Int64;
//Euclids subtraction method (http://en.wikipedia.org/wiki/Euclidean_algorithm)
//only workswith positive integers
begin
if (a = 0) then Exit(b);
if (b = 0) then Exit(a);
if (a < 0) then a := -a;
if (b < 0) then b := -b;
while not (a = b) do
begin
if (a > b) then
a := a - b
else
b := b - a;
end;
Result := a;
end;
procedure AdjustPrecision(var Precision: Double; Value: Double);
const
MaxPrec = Double(1.0)/MaxInt64;
begin
Precision := Abs(Precision);
if ((Abs(Value) / Precision) > 1E15) then
begin
//writeln('Value / Precision > 1E15');
Precision := Abs(Value) / 1E16;
end;
if (Precision < MaxPrec) then
Precision := MaxPrec;
end;
function InRange64(Value: Double): Boolean; inline;
begin
Result := not ((Value > MaxInt64) or (Value < MinInt64));
end;
procedure CheckRange(Value: Double);
begin
if not InRange64(Value) then
raise ERangeError.Create(SRangeCheckError);
end;
function IsBorderlineValue(Value: Double; out F: TFraction): Boolean;
const
MaxPrec = Double(1.0)/MaxInt64;
ZeroBoundary = MaxPrec / 2;
begin
if (Abs(Value) <= MaxPrec) then
begin
Result := True;
//writeln('Abs(Value) < 1/MaxInt64 [',MaxPrec,']');
if (Abs(Value) < ZeroBoundary) then
begin
//writeln('Abs(Value) < ZeroBoundary [',ZeroBoundary,']');
F.Numerator := 0;
F.Denominator := 1;
end
else
begin
if (Value < 0) then
F.Numerator := -1
else
F.Numerator := 1;
F.Denominator := MaxInt64;
end;
end
else
Result := False;
end;
//uses method of Continued fractions
function CF_FloatToFraction(Value, Precision: Double): TFraction;
var
H1, H2, K1, K2, A, NewA, tmp: Int64;
B, diff, test: Double;
PendingOverFlow, Found: Boolean;
begin
CheckRange(Value);
AdjustPrecision(Precision, Value);
//Borderline cases
if IsBorderlineValue(Value, Result) then
Exit;
H1 := 1;
H2 := 0;
K1 := 0;
K2 := 1;
b := Value;
NewA := Round(Floor(b));
repeat
A := NewA;
tmp := H1;
H1 := (a * H1) + H2;
H2 := tmp;
tmp := K1;
K1 := (a * K1) + K2;
K2 := tmp;
//write('H1=',H1,' K1=',K1,' A=',A);
test := H1 / K1;
//write(' test=',test);
diff := Abs(test - Value);
//write(' diff=',diff);
Found := (diff < Precision);
if not Found then
begin
if (Abs(B-A) < 1E-30) then
B := 1E30 //happens when H1/K2 exactly matches Value
else
B := 1 / (B - A);
//write(' B=',B);
PendingOverFlow := (((Double(B) * H1) + H2) > MaxInt64) or
(((Double(B) * K1) + K2) > MaxInt64) or
(B > MaxInt64);
//writeln(' PendingOverFlow=',PendingOverFlow);
if not PendingOverFlow then
NewA := Round(Floor(B));
{
if PendingOverFlow then
begin
writeln('PendingOverFlow');
writeln('New H1 = ',(Double(NewA) * H1) + H2);
writeln('New K1 = ',(Double(NewA) * K1) + K2);
writeln('B = ',B);
writeln('MaxInt64 = ',Double(MaxInt64));
end;
}
end;
until Found or PendingOverFlow;
Result.Numerator := H1;
Result.Denominator := K1;
end;
{
This implementation of FloatToFraction was originally written by:
David Peterson and The Math Forum @ Drexel
Source: http://mathforum.org/library/drmath/view/51886.html
It was ported to FreePascal by Bart Broersma
Adjustments made:
* Precision is bound by a magnitude of -15 to Value
* Bordeline cases close to zero are handled
* Handle negative values
* Handle negative precision
* Original code dealt with 32-bit integers, it was adjusted for 64-bit integers
The original copyrigh holder has granted me permission to adjust and redistribute
this code under LGPL license.
When redistributing this code, the comments above MUST also be redistributed with it!
}
function MF_FloatToFraction(Value, Precision: Double): TFraction;
var
IntPart, Count, Num, Denom: Int64;
i: Integer;
TestLow, TestHigh, Test: Double;
L,H: TFraction;
IsNeg: Boolean;
begin // find nearest fraction
//writeln('MF_FloatToFraction:');
//writeln(' Value = ',Value);
//writeln(' Prec = ',Precision);
CheckRange(Value);
AdjustPrecision(Precision, Value);
//Borderline cases
if IsBorderlineValue(Value, Result) then
Exit;
IsNeg := (Value < 0);
Value := Abs(Value);
intPart := Round(Int(Value));
Value := Frac(Value);
L.Numerator := 0;
L.Denominator := 1;
H.Numerator := 1;
H.denominator := 1;
for i := 1 to 100 do //was 100
begin
//writeln(' i = ',i);
testLow := L.Denominator * Value - L.Numerator;
testHigh := H.Numerator - H.Denominator * Value;
//if (testHigh < Precision * H.Denominator) then
if (Abs(H.ToFloat - Value) < Precision) then
begin
//writeln(' (testHigh < Precision * H.Denominator)');
//writeln(' testHigh = ',testHigh);
//writeln(' Precision * H.Denominator = ',Precision * H.Denominator);
break; // high is answer
end;
//if (testLow < Precision * L.Denominator) then
if (Abs(L.ToFloat - Value) < Precision) then
begin // low is answer
//writeln(' (testLow < Precision * L.Denominator)');
//writeln(' testLow = ',testLow);
//writeln(' Precision * L.Denominator = ',Precision * L.Denominator);
H := L;
break;
end;
if Odd(i) then
begin // odd step: add multiple of low to high
//writeln(' Odd step');
test := testHigh / testLow;
count := Round(Int(test)); // "N"
num := (count + 1) * L.Numerator + H.Numerator;
denom := (count + 1) * L.Denominator + H.Denominator;
if ((num > High(Int64) - 1) or // was 8000, 10000
(denom > High(Int64) - 1)) then
//if ((num > $8000) or // was 8000, 10000
//(denom > $10000)) then
begin
//writeln(' ((num > High(Int64) - 1) or (denom > High(Int64) - 1))');
break;
end;
H.Numerator := num - L.Numerator; // new "A"
H.Denominator := denom - L.Denominator;
L.Numerator := num; // new "B"
L.Denominator := denom;
end
else
begin // even step: add multiple of high to low
//writeln(' Even step');
test := testLow / testHigh;
count := Round(Int(test)); // "N"
num := L.Numerator + (count + 1) * H.Numerator;
denom := L.Denominator + (count + 1) * H.Denominator;
if ((num > High(Int64) - 1) or //10000. 10000
(denom > High(Int64) - 1)) then
//if ((num > $10000) or //10000. 10000
//(denom > $10000)) then
begin
//writeln(' ((num > High(Int64) - 1) or (denom > High(Int64) - 1))');
break;
end;
L.Numerator := num - H.Numerator; // new "A"
L.Denominator := denom - H.Denominator;
H.Numerator := num; // new "B"
H.Denominator := denom;
end;
//writeln(' End of loop iteration for i = ',i);
end;
//return Fraction(intPart, 1) + high;
//writeln('MF_FloatToFraction: assigning Result');
//writeln('H = ',H.ToString);
//writeln('IntPart := ',IntPart);
//Avoid call to TFraction.Normalize in Result := H + IntPart
Result := H;
Result.Numerator := Result.Numerator+ (Result.Numerator * IntPart);
if IsNeg then
Result.Numerator := -Result.Numerator;
//writeln('MF_FloatToFraction End.');
end;
function TryFloatToFraction(Value, Precision: Double; out F: TFraction;
AcceptPrecisionError: Boolean): Boolean;
begin
Result := False;
if not InRange64(Value) then Exit;
AdjustPrecision(Precision, Value);
try
F := FloatToFraction(Value, Precision);
Result := AcceptPrecisionError or (Abs(Value - F.ToFloat) <= Precision)
except
Result := False;
end
end;
function FloatToFractionDef(Value, Precision: Double; Def: TFraction;
AcceptPrecisionError: Boolean): TFraction;
begin
if not TryFloatToFraction(Value, Precision, Result, AcceptPrecisionError) then
Result := Def;
end;
function StrToFraction(const S: String): TFraction;
begin
if not TryStrToFraction(S, Result) then raise EConvertError.CreateFmt(SInvalidFraction, [S]);
end;
{
S is either:
A single fraction e.g. 2/3
A single integer e.g. 10
An integer + a fraction, separated by a single space (#32) e.g. 2 1/2 (two and a half)
}
function TryStrToFraction(const S: String; out F: TFraction): Boolean;
type
TFracType = (ftComplex, ftSingleFraction, ftInteger);
var
FracType: TFracType;
SInt, SNum, SDen: String;
IntPart, NumPart, DenPart: Integer;
PSpace, PFrac: SizeInt;
begin
Result := False;
SInt := '';
SNum := '';
SDen := '';
IntPart := 0;
NumPart := 0;
DenPart := 0;
PSpace := Pos(#32, S);
PFrac := Pos(FracSymbol,S);
if (PSpace > 0) and (PFrac > 0) then FracType := ftComplex
else if (PSpace = 0) and (PFrac > 0) then FracType := ftSingleFraction
else FracType := ftInteger;
case FracType of
ftComplex:
begin
//writeln('ftComplex');
SInt := Copy(S, 1, PSpace - 1);
SNum := Copy(S, PSpace + 1, PFrac - PSpace - 1);
SDen := Copy(S, PFrac + Length(FracSymbol), MaxInt);
//no spaces allowed in SNum or SDen
//writeln('SInt = "',sint,'" SNum = "',snum,'" SDen = "',sden,'"');
if (Pos(#32, SNum) > 0) or (Pos(#32, SDen) > 0) then Exit;
if not (TryStrToInt(SInt, IntPart) and TryStrToInt(SNum, NumPart) and TryStrToInt(SDen, DenPart))
or (DenPart = 0) or (NumPart < 0) then
Exit;
if (DenPart < 0) then
begin
DenPart := -DenPart;
NumPart := -NumPart;
end;
if (IntPart = 0) then
begin
F.Numerator := NumPart;
F.Denominator := DenPart;
end
else
begin
if (NumPart < 0) then
begin
NumPart := -NumPart;
IntPart := -IntPart;
end;
if (IntPart > 0) then
F.Numerator := NumPart + Int64(DenPart) * IntPart
else
F.Numerator := -(NumPart - Int64(DenPart) * IntPart);
F.Denominator := DenPart;
end;
end;
ftSingleFraction:
begin
//writeln('ftSingleFraction');
SNum := Copy(S, 1, PFrac - 1);
SDen := Copy(S, PFrac + Length(FracSymbol), MaxInt);
//writeln('SNum = "',snum,'" SDen = "',sden,'"');
//no spaces allowed in SNum or SDen
if (Pos(#32, SNum) > 0) or (Pos(#32, SDen) > 0) then Exit;
if not (TryStrToInt(SNum, NumPart) and TryStrToInt(SDen, DenPart))
or (DenPart = 0) then
Exit;
F.Numerator := NumPart;
F.Denominator := DenPart;
end;
ftInteger:
begin
//writeln('ftInteger');
//no spaces allowed in SInt
if (Pos(#32, SInt) > 0) then Exit;
if not TryStrToInt(S, IntPart) then Exit;
F.Numerator := IntPart;
F.Denominator := 1;
end;
end;//case
Result := True;
end;
function StrToFractionDef(const S: String; Def: TFraction): TFraction;
begin
if not TryStrToFraction(S, Result) then Result := Def;
end;
function Min(a, b: TFraction): TFraction;
begin
if (a < b) then
Result := b
else
Result := b;
end;
function Max(a, b: TFraction): TFraction;
begin
if (a > b) then
Result := a
else
Result := b;
end;
function InRange(const AValue, AMin, AMax: TFraction): Boolean;
begin
Result := (AValue >= AMin) and (AValue <= AMax);
end;
function EnsureRange(const AValue, AMin, AMax: TFraction): TFraction;
begin
If (AValue < AMin) then
Result := AMin
else if (AValue > AMax) then
Result := AMax
else
Result := AValue;
end;
function Sign(const AValue: TFraction): TValueSign;
begin
if (AValue.Denominator < 0) then
begin
AValue.Numerator := -AValue.Numerator;
AValue.Denominator := -AValue.Denominator;
end;
if (AValue.Numerator < 0) then
Result := NegativeValue
else if (AValue.Numerator > 0) then
Result := PositiveValue
else
Result := ZeroValue;
end;
function IsZero(const AValue: TFraction): Boolean;
begin
Result := (AValue.Numerator = 0);
end;
function Abs(const AValue: TFraction): TFraction;
begin
Result := AValue;
Result.Normalize;
if (Result.Numerator < 0) then Result.Numerator := -Result.Numerator;
end;
function Fraction(ANumerator, ADenominator: Int64): TFraction;
begin
Result.Numerator := ANumerator;
//will raise exception if ADenominator = 0
Result.Denominator := ADenominator;
end;
function Fraction(AIntPart, ANumerator, ADenominator: Int64): TFraction;
var
IsNeg: Boolean;
begin
if (ANumerator < 0) then
raise EMathError.CreateFmt(SInvalidFraction,[Format('%d %d/%d',[AIntPart,ANumerator,ADenominator])]);
IsNeg := (AIntPart < 0) xor (ADenominator < 0);
AIntPart := Abs(AIntPart);
ADenominator := Abs(ADenominator);
Result.Numerator := ANumerator + (ADenominator * AIntPart);
if IsNeg then Result.Numerator := - Result.Numerator;
//will raise exception if ADenominator = 0
Result.Denominator := ADenominator;
end;
operator = (F1: TFraction; F2: TFraction) B: Boolean;
begin
F1.Normalize;
F2.Normalize;
B := (F1.Numerator = F2.Numerator) and (F2.Denominator = F2.Denominator);
end;
operator < (F1: TFraction; F2: TFraction) B: Boolean;
begin
F1.Normalize;
F2.Normalize;
B := (F1.Numerator * F2.Denominator) < (F2.Numerator * F1.Denominator);
end;
operator > (F1: TFraction; F2: TFraction) B: Boolean;
begin
F1.Normalize;
F2.Normalize;
B := (F1.Numerator * F2.Denominator) > (F2.Numerator * F1.Denominator);
end;
operator <= (F1: TFraction; F2: TFraction) B: Boolean;
begin
B := (F1 < F2) or (F1 = F2);
end;
operator >= (F1: TFraction; F2: TFraction) B: Boolean;
begin
B := (F1 > F2) or (F1 = F2);
end;
operator := (I: Int64) F: TFraction;
begin
F.Numerator := I;
F.Denominator := 1;
end;
operator := (S: String) F: TFraction;
begin
if not TryStrToFraction(S, F) then
raise EConvertError.CreateFmt(SInvalidFraction, [S]);
end;
operator + (L: TFraction; R: TFraction) F: TFraction;
begin
F.Numerator := L.Numerator * R.Denominator + R.Numerator * L.Denominator;
F.Denominator := L.Denominator * R.Denominator;
F.Normalize;
end;
operator - (L: TFraction; R: TFraction) F: TFraction;
begin
R.Numerator := - R.Numerator;
F := L + R;
end;
operator - (L: TFraction) F: TFraction;
begin
F.Numerator := - L.Numerator;
F.Denominator := L.Denominator;
F.Normalize;
end;
operator * (L: TFraction; R: TFraction) F: TFraction;
begin
L.Normalize;
R.Normalize;
F.Numerator := L.Numerator * R.Numerator;
F.Denominator := L.Denominator * R.Denominator;
F.Normalize;
end;
operator * (L: TFraction; R: Int64) F: TFraction;
begin
F := L;
F.Normalize;
F.Numerator := L.Numerator * R;
F.Normalize;
end;
operator * (L: Int64; R: TFraction) F: TFraction;
begin
F := R * L;
end;
operator / (L: TFraction; R: TFraction) F: TFraction;
var
Temp: TFraction;
begin
Temp.Numerator := R.Denominator;
//this will raise an exception if R = 0
Temp.Denominator := R.Numerator;
F := L * Temp;
end;
operator / (L: TFraction; R: Int64) F: TFraction;
begin
if (R = 0) or (L.Denominator = 0) then raise EZeroDivide.Create(SDivisionByZero);
F := L;
F.Normalize;
F.Denominator := F.Denominator * R;
F.Normalize;
end;
operator ** (L: TFraction; R: Integer) F: TFraction;
var
i: Integer;
begin
L.Normalize;
F := L;
if (R = 0) then
begin
if (L.Numerator = 0) and not AllowZeroPowerZero then raise EMathError.Create(SZeroPowerZero);
F.Numerator := 1;
F.Denominator := 1;
end
else if (R > 0) then
begin
for i := 1 to R-1 do F := F * L
end
else
begin
F := 1 / F;
L := F;
for i := 1 to -(R)-1 do F := F * L ;
end;
end;
procedure TFraction.SetDominator(AValue: Int64);
begin
if (AValue = 0) then raise EZeroDivide.Create(SDivisionByZero);
if (FDenominator = AValue) then Exit;
FDenominator := AValue;
end;
procedure TFraction.Normalize;
var
GCD: Int64;
begin
if (Denominator < 0) then
begin
Numerator := - Numerator;
Denominator := - Denominator;
end;
GCD := GreatestCommonDivisor(Numerator, Denominator);
if (GCD <> 1) then
begin
Numerator := Numerator div GCD;
Denominator := Denominator div GCD
end;
end;
function TFraction.ToString: String;
begin
if (Denominator < 0) then
begin
Numerator := - Numerator;
Denominator := - Denominator;
end;
Result := IntToStr(Numerator) + FracSymbol + IntToStr(Denominator);
end;
function TFraction.Resolve: String;
var
IntPart: Int64;
begin
Normalize;
if (Abs(Numerator) > Abs(Denominator)) then
begin
IntPart := Numerator div Denominator;
Numerator := Numerator mod Denominator;
if (IntPart < 0) then Numerator := -Numerator;
if (Numerator <> 0) then
Result := IntToStr(IntPart) + #32 + ToString
else
Result := IntToStr(IntPart);
end
else
begin
Result := ToString;
end;
end;
function TFraction.ToFloat: Double;
begin
Result := Numerator / Denominator;
end;
end.

70
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From: r!chard tchen <richard@mathforum.org>
To: Bart Broersma <bartjunk64@gmail.com>
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Subject: Re: Can I use your code in my library?
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On Sun, 22 Mar 2015, Bart Broersma wrote:
> forumURL: http://mathforum.org/library/drmath/view/51886.html
> message: Hi,
>
> I ported the C++ code on this page (Decimal To Fraction Conversion) to Pascal.
>
> I would like to add this to my Fractions library.
> This library is modified LGPL (with linking exception).
>
> Would you grant me permission to do so?
> In the long run the library may become part of the FreePascal component library.
> (FreePascal is an open souce Pascal compiler, which is used for the Lazarus IDE, for which I am one of the devolopers: see http://www.lazarus-ide.org/ for more info on this project.)
>
> Sincerely yours,
>
> Bart Broersma
Dear Bart,
Thanks for writing the Math Forum @ Drexel.
You're welcome to take the C++ code from that Ask Dr. Math conversation and add it to your Pascal library. Please just give us proper attribution (in comments?) and clearly identify that code as a resource of David Peterson and The Math Forum @ Drexel, and not something of your own creation.
On this reply, Bart, I've "Bcc:"ed Doctor Peterson, so that he may appreciate your interest.
Let us know how this helps you, FreePascal, and the greater Lazarus community.
--
Richard Tchen
The Math Forum @ Drexel
3210 Cherry Street
Philadelphia, PA 19104-2713
tel1: (215)895-1787
tel2: (800)756-7823
facs: (215)895-2964
The Math Forum @ Drexel (http://mathforum.org/)
is a research and educational enterprise of
The Drexel University School of Education (http://goodwin.drexel.edu/soe/).